Kalman equalizer

ABSTRACT

An improved Kalman equalizer for providing a tap weight of a transversal filter for equalizing a transmission line has been found. A complex number fast Kalman equalizer (FIG. 27), which is the improvement of the prior fast real number Kalman equalizer (see; &#34;Fast calculation of gain matrices for recursive estimation schemes&#34; by Lennart Ljung, et al, INT. J. Control, 1978, vol. 27, No. 1, 1-19), has a complex conjugate generator (C 2 ) for processing a complex input signal so that an equalization can be performed on the bandwidth division concept, and a transmission line having a complex impulse response can be equalized. Then, a bandwidth division equalization in which a whole bandwidth is separated into a plurality of sub-bandwidth each being equalized by a related sub-equalizer becomes possible. Also, a QAM signal (Quadrature Amplitude Modulation) which has a complex impulse response can be equalized. A bandwidth division fast Kalman equalizer (FIG. 68) is also possible as the modification of the present invention.

BACKGROUND OF THE INVENTION

This invention is concerned with automatic equalizer for data transmission having less calculation amount and faster and stabler convergence characteristics.

Formerly, there were two types of automatic equalizers, one of which is a learn and identification type equalizer or ordinary equalizer having slow and unstable convergence and the other type of which is a fast Kalman equalizer for AM modulation systems having fast and stable convergence. However, these have shortcomings, namely, slow and unstable convergence, large amount of calculating labour or restriction of usage to AM modulation, respectively.

Further, prior Kalman equalizers have the disadvantage that only real number signals are handled, but complex number signals can not be handled.

SUMMARY OF THE INVENTION

The general purpose of the present invention is to overcome said disadvantages of prior equalizers.

The present invention is aimed to provide a bandwidth division type Kalman equalizer in which a hardware amount for the calculation is reduced, the operation is stable, and the faster convergence characteristics are obtained.

Another object of the present invention is to provide a Kalman type equalizer, in which a complex number is dealt with, and a QAM modulation signal is also dealt with.

According to the present invention, a fast Kalman equalizer having a complex impulse response for equalizing a complex input signal comprises;

(a) an input terminal Q₅ for receiving an input signal g_(k),

(b) a transversal filter (TFQ) having at least a tapped delay line (TDLQ₁) with a plurality of taps and an adder (SUMQ₁) for summing up all the tap outputs of said tapped delay line (TDLQ₁),

(c) an output terminal Q₇ coupled with the output of said adder (SUMQ₁) for providing an equalized output signal,

(d) a tap gain memory (RQ) for providing a tap coefficient to said tapped delay line (TDLQ₁), by accepting the increment of the tap weight using the formula Δh_(k) =k_(k) e_(k) (Eq. 12),

(e) a multiplicator (MQ₂₂) for providing the product of the reference signal e_(k) from a reference terminal (Q₆) and k_(k) (Eq. 14),

(f) an A-register control (PTAQ) for providing the values q_(k) (Eq. 18) and A_(k) (Eq. 18') according to the vectors x_(k) (Eq. 9), x_(k+1) (Eq. 16), k_(k), and A_(k-1),

(g) an S-register control (PTSQ) for providing (a) v_(k) which is the complex conjugate of v_(k) by the generator (C₂), said v_(k) is obtained by Eq. 19 (v_(k) =g_(k) +A_(k) ^(T) ·x_(k), where A_(k) ^(T) is the transposed matrix of the matrix A_(k)), and (b) S_(k) =S_(k-1) +v_(k) g_(k) (Eq.20'), according to the values A_(k), x_(k), and g_(k),

(h) a D-register control (PTDQ) for providing D_(k) (Eq. 25), m_(k) (Eq. 22 and Eq. 21), and w_(k) (Eq. 23),

(i) a k-register control (PTkQ) for providing k_(k+1) by the formula; k_(k+1) =m_(k) -D_(k) w_(k) (Eq. 26),

(j) means for providing the value k_(k+1) obtained in said item (i) to the multiplicator (MQ₂₂) of the item (e), the A-register control (PTAQ) of the item (f), and the D-register control (PTDQ) of the item (h), for the next step calculation.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, features, and attendant advantages of the present invention will be appreciated as the same become better understood by means of the following description and accompanying drawings wherein;

FIG. 1 is a block diagram of AM modulation data transmission system,

FIG. 2 is a figure of equivalent circuit for AM modulation data transmission system,

FIG. 3 is a block diagram of QAM modulation data transmission system,

FIG. 4 is a figure of equivalent circuit for QAM modulation data transmission system,

FIG. 5 is a block diagram of real number Kalman equalizer,

FIG. 6 is a detailed figure for real number Kalman equalizer,

FIG. 7 is a block diagram of complex number Kalman equalizer,

FIG. 8 is a detailed figure for complex number Kalman equalizer,

FIG. 9 is a detailed figure of compound type expression of complex number tapped delay lines TDLQ₁ and TDLQ₂,

FIG. 10 is a detailed figure of separate type expression of complex number tapped delay lines TDLQ₁ and TDLQ₂,

FIG. 11 is a detailed figure for complex number multiplier MQ₁,

FIG. 12 is a detailed figure of separate type expression for complex number multiplier MQ₁₁,

FIG. 13 is a detailed figure of separate type expression for complex number concentrator SUMQ₁,

FIG. 14 is a detailed figure of separate type expression for complex number resister RQ,

FIG. 15 is a detailed figure of separate type expression for complex conjugate generator C,

FIG. 16 is a detailed figure for complex number P resister PQ,

FIG. 17 is a detailed figure for complex number multiplier MQ₂,

FIG. 18 is a detailed figure for combined portion of complex number multiplier MQ₃ and complex number concentrator SUMQ₂,

FIG. 19 is a detailed figure for complex number multiplier MQ₄,

FIG. 20 is a detailed figure for complex number multiplier MQ₆,

FIG. 21 is a detailed figure for complex number multiplier MQ₇,

FIG. 22 is a detailed figure for unit matrix generator U,

FIG. 23 is a detailed figure for complex number multiplier MQ₈,

FIG. 24 is a block diagram for real number fast Kalman equalizer,

FIG. 25 is a detailed figure for real number fast Kalman equalizer,

FIG. 26 is a block diagram for complex number fast Kalman equalizer,

FIG. 27 is a detailed figure for complex number fast Kalman equalizer,

FIG. 28 is a detailed figure for complex number delay elements DLQ₁ through DLQ₄,

FIG. 29 is a detailed figure for complex number tapped delay line TDLQ₁,

FIG. 30 is a detailed figure for complex number tapped delay line TDLQ₂,

FIG. 31 is a detailed figure for complex number multiplier MQ₁,

FIG. 32 is a detailed figure for complex number concentrator SUMQ₁,

FIG. 33 is a detailed figure for complex number resister RQ,

FIG. 34 is a detailed figure for complex number dimension increaser DIQ₁,

FIG. 35 is a detailed figure for complex number dimension increaser DIQ₂,

FIG. 36 is a detailed figure for complex number multiplier MQ₁₁,

FIG. 37 is a detailed figure for complex number multiplier MQ₁₂,

FIG. 38 is a detailed figure for complex number adder ADQ₁₂,

FIG. 39 is a detailed figure for complex number A resister ARQ,

FIG. 40 is a detailed figure for complex number multiplier MQ₁₄,

FIG. 41 is a detailed figure of separate type expression for complex conjugate generator C₂,

FIG. 42 is a detailed figure for complex number inverse value generator INQ₁,

FIG. 43 is a detailed figure for complex number multiplier MQ₁₆,

FIG. 44 is a detailed figure for complex number adder ADQ₁₅,

FIG. 45 is a detailed figure for complex number dimension increaser DIQ₃,

FIG. 46 is a detailed figure for complex number dimension decreaser DDQ₁,

FIG. 47 is a detailed figure for complex number multiplier MQ₁₇,

FIG. 48 is a detailed figure for complex number multiplier MQ₁₉,

FIG. 49 is a detailed figure for complex number multiplier MQ₂₀,

FIG. 50 is a detailed figure for complex number adder ADQ₁₈,

FIG. 51 is a detailed figure for complex number D resister DRQ,

FIG. 52 is a detailed figure for complex number multiplier MQ₂₁,

FIG. 53 is a detailed figure for complex number adder ADQ₁₉,

FIG. 54 is a detailed figure for complex number k resister kRQ,

FIG. 55 is a figure for portion around equalizer extracted from equivalent circuit for QAM modulation data transmission system,

FIG. 56 is a modification of FIG. 55 in which complex number fast Kalman equalizer KEQQ is used as complex number equalizer EQQ,

FIG. 57 is a figure obtained by introduction of band pass filter directly after the input point of complex information value,

FIG. 58 is a figure obtained by modification of the configuration in FIG. 57 taking advantage of bandwidth restriction of signal in the configuration,

FIG. 59 is a figure reduced from composition in FIG. 58 by moving the position of band pass filter,

FIG. 60 is the figure showing a construction method of bandwidth division Kalman equalizer,

FIGS. 61(a) through (d) are figures of characteristics of band pass filters used in the equalizer in FIG. 60,

FIG. 62 is the figure for explanation of operation of bandwidth Kalman equalizer laying emphasis on signal path,

FIG. 63 is the figure for illustration of correctness of convergence at bandwidth division equalizer,

FIG. 64 is a figure for another type of bandwidth division Kalman equalizer,

FIGS. 65(a) through (c) are figures of characteristics of band pass filters used in the equalizer in FIG. 64,

FIGS. 66(a) through (c) are figures continued from FIG. 65,

FIG. 67 is the figure showing the construction way of band pass filter utilizing configuration of transversal filter, and

FIG. 68 is a figure for bandwidth division fast Kalman equalizer.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

First, some background idea of the present invention will be described for the easy understanding of the invention.

FIG. 1 is a block diagram of an AM modulation data transmission system. In FIG. 1, P₁, P₂, P₃, P₄, P₅, P₆, P₇, and P₈ are input point of information symbol, output point of transmission terminal equipment, input point of receiving terminal equipment, input point of sampler, input point of automatic equalizer, input point of reference signal, output point of automatic equalizer and output point of decision circuit, respectively. Furthermore BSG₁, AMM₁, CG₁, CG₂, TRL, AMD₁, S₁, EQA, DEC, and DL₁ are base band signal generator, AM modulator, carrier generator at transmission equipment terminal, carrier generator at receiving equipment terminal, transmission line, AM demodulator, sampler, automatic equalizer, decision circuit and imaginary delay line explained later. Information symbol value is generated in T second interval, which is called as symbol sending interval of this transmission system, at point P₁. The values of information symbols are chosen among the formerly prescribed values and carry information. Time series of information symbol values passes through the base band generator BSG₁ , which operates as low pass filter and which makes the modulating operation at modulator AMM₁ easier, and consecutively introduced in AM modulator AMM₁ and is modulated there. This AM modulator is controlled by carrier generator CG₁ generating carrier expressed as cos ω_(c) t. This means the frequency of the carrier is ω_(c) (radian/second). Output signal of AM modulator passes through output point of transmission terminal equipment, enters in transmission line TRL and appears at input point of receiving terminal equipment. This signal then enters at AM demodulator AMD₁ and is demodulated. CG₂ is a carrier generator operating in synchronization with the carrier generator CG₁ at the transmission equipment terminal. Output signal of AMD₁ is applied, at input point of sampler P₄, to sampler S₁ which samples incoming signal at T second interval. The output signal of the sampler is generated at P₅ which is input point of automatic equalizer. The automatic equalizer operates under the control of input signal to automatic equalizer generated at P₅ and reference signal generated at P₆. The parameters at automatic equalizer is adjusted so that output value of equalizer becomes as near as possible to the reference signal applied at P₆. The above mentioned reference signal is chosen to be the information signal value at P₁ but affected under the effect of a certain amount of delay caused by imaginary delay line DL₁. Equalizer output signal generated at point P₇ is introduced to decision circuit which eliminates noise signal and produces correct information signal value corresponding one of the formerly prescribed information values. Thus communication information is finally extracted. The dotted line and imaginary delay line DL₁ in FIG. 1 are added to explain the nature of reference signal which is input information symbol delayed at DL₁. These parts act as equivalent circuit in the figure. At actual transmission system, reference signal generated at P₅ is obtained as output signal of decision circuit DEC given at P₈ or obtained by producing it independently at receiving terminal equipment when transmitted symbol sequence is known. The above mentioned thing is situation for actual data transmission system. Components BSG₁, AMM₁, CG₁, TRL, AMD₁, and CG₂ can be replaced by equivalent base band transmission line ETRLA as shown in FIG. 2. The operation of automatic equalizer can be expressed easier through the use of this equivalent circuit. Other parts, in this case, can be regarded as unchanged. Following these lines of consideration, it will be easily understood that automatic equalizer EQA provides inverse characteristics of equivalent base band transmission line ETRLA and that the cascade connection of ETRLA and automatic equalizer EQA must be reduced to be a delay line with the same delay given in delay line DL₁. The configuration of FIGS. 1 and 2 can deal only with real number values including information symbol values generated at P₁. As AM modulation data transmission system accepts only real number, this type of system is called as real number data transmission system and automatic equalizer EQA in this system is called real number automatic equalizer.

QAM modulation system will be shown next. FIG. 3 is a block diagram of QAM modulation data transmission system. Q₁ R and Q₁ I, in this figure, are input point of real number information symbol and input point of imaginary number information symbol respectively and these two points are combined into input point of information symbol Q₁. Points Q₂, Q₃, Q₄, Q₅, Q₇ and Q₈ are output point of transmission terminal equipment, input point of receiving terminal equipment, input point of sampler, input point of automatic equalizer, input point of reference signal, output point of automatic equalizer and output point of decision circuit, respectively. BSGR and BSGI are a base band signal generator for real number and a base band signal generator for imaginary number, respectively. These have characteristics of low pass filter. QMMR and QMMI are QAM modulator for real number and QAM modulator for imaginary number, respectively, and the component consisted of these two circuits is called QAM modulator QMM. CGR₁ is carrier generator for real number at transmission terminal equipment which generates carrier wave cos ω_(c) t with angle frequency of ω_(c) (radian/second) and controls aforementioned QAM modulator for real number QAMMR. CGI₁ is also carrier generator for imaginary number which generates carrier wave sin ω_(c) t and controls QMMI. SUM₁ is an adder which simply adds its two inputs. TRL is, as formerly explained, transmission line. QMDR is QAM demodulator for real number and QMDI is QAM demodulator for imaginary number and these two parts are combined into QAM demodulator QMD. CGR₂ and CGI₂ are carrier generators at receiving terminal equipment and generate, in synchronization with CGR₁ and CGRI₁ respectively at the transmission terminal equipment, carrier waves cos ω_(c) t and sin ω_(c) t respectively and drives QMDR and QMDI respectively. M₁ is imaginary number unit multiplier and multiplies imaginary number unit j to input providing the result at its output. SUM₂ and S₂ are an adder and a sampler, respectively. EQQ is QAM automatic equalizer called complex number automatic equalizer as a result of its ability to deal with complex number. DEC₂ is decision circuit. M₂ is, as M₁, imaginary unit multiplier. SUM₃ is adder. DL₁ and DL₂ are delay lines with a certain amount of delay providing the same effect as that of DL₁ in FIG. 1.

Next, the operation of QAM data transmission system in FIG. 3 will be explained. At Q₁ R and Q₁ I real number information symbol and imaginary number information symbol are generated respectively with the interval of T second which is symbol sending interval of this transmission system. The values generated must be chosen from formerly prescribed set of values. Imaginary number information symbols, of course, must take real number for their values and not imaginary numbers in the actual transmission system. The reason why these values are called imaginary will be explained later. The time series of information symbols generated at Q₁ R and Q₁ I respectively are introduced to base band signal generator for real number BSGR and base band signal generator for imaginary number BSGI respectively and consecutively modulated by modulators QMMR and QMMI respectively. It must be noticed that carriers which are generated at CGR₁ and CGI₁ respectively and which control two modulators are different in phase by 90 degree from each other. Output signals from QMMR and QMMI are summed up at SUM₁ and sent to transmission line TRL through the output point of transmission terminal equipment Q₂. Output signal of transmission line appears at input point of receiving terminal equipment Q₃. The signal is led to QAM demodulator for real number QMDR and QAM demodulator for imaginary number QMDI respectively both of them conducting QAM demodulation operation. Two carrier generators CGR₂ and CGI₂ which operate in synchronization with carrier generators at the transmission terminal equipment generate carrier waves cos ω_(c) t and sin ω_(c) t respectively. To the output signal of QMDI, imaginary number unit j is multiplied at the imaginary unit multiplier M₁ and the result is added with output signal of QMDR at the adder SUM₂. Output value of SUM₂ is introduced to sampler S₂ which samples its input signal at T second interval. The output of the sampler appears at the input point of automatic equalizer Q₅. Complex number automatic equalizer EQQ operates in accordance with input signal of automatic equalizer and reference signal at Q₆ adjusting parameters in the automatic equalizer so that output signal of the complex number automatic equalizer reaches as near as possible to the reference signal. The component expressed in dotted line is to show that the reference signal generated at Q₆ is the value given as the output signal of adder SUM₃ which admits two signals as its input, one of which is value at the input point of real number information symbol Q₁ R delayed by delay line DL₁ for a certain amount of delay, and the other of which is value at the input point of imaginary number information symbol Q₁ I multiplied by imaginary unit j at the imaginary number unit multiplier M₂ and delayed by DL₂ having the same delay of DL₁. The reference signal will be, as explained at the part of AM modulation, often made at receiving terminal equipment. Equalizer output at point Q₇ is introduced to decision circuit DEC₂, where noise is eliminated giving rise to re-generation of information symbols which take values from prescribed set of values. These are situations for actual data transmission system. BSGR, BSGI, QMMR, QMMI, CGR₁, CGI₁, SUM₁, TRL, QMDR, QMDI, CGR₂, CGI₂ and SUM₂ can be replaced by and equivalent circuit called complex number equivalent base band transmission line ETRLQ as shown in FIG. 4. The operation of equalizer can be understood easier using this equivalent circuit. The complex number equivalent base band transmission line ETRLQ has complex number values as its impulse response. In the case generation part of input information symbol and reference signal can be combined as shown in FIG. 4. In FIG. 4, M₃ is imaginary number unit multiplier, and DL₃ is complex number delay line, and SUM₄ is adder. Input signal to complex number equivalent base band transmission line ETRLQ can be obtained as output signal of adder SUM₄ and called complex number information symbol which becomes to be reference signal after passing through complex number delay line DL₃. The adder SUM₄ admits two inputs, one of which is real number information symbol introduced at Q₁ R and the other of which is imaginary number information symbol introduced at Q₁ I multiplied by imaginary number unit at M₃. Under this situation, complex number automatic equalizer EQQ must provide inverse transmission function of complex number equivalent base band transmission line ETRLQ and cascade connection of EQQ and ETRLQ can be replaced by a single delay line having the same amount of delay as that of DL₃. As shown above, the system shown in FIG. 4 can deal with complex values as exemplified with its complex number information symbol and complex number equivalent base band transmission line having impulse response of complex number. Consequently this kind of QAM modulation data transmission system is called as complex number data transmission system and the equalizer EQQ in the system is of course called complex number automatic equalizer.

The role of automatic equalizers are explained and configurations and operation methods of automatic equalizers are shown so far. Henceforth configuration and operation of automatic equalizers will be treated. First, two kinds of systems which are composing elements of this invention will be dealt with. They are real number Kalman equalizer and complex number Kalman equalizer. At the beginning, real number Kalman equalizer will be discussed. Composing method of EQA in FIGS. 1 and 2 which is called real number automatic equalizer using real number Kalman equalizer will be shown. Inner structure of EQA when composed by real number Kalman equalizer is divided into large sections as shown in FIG. 5. KEQA is real number Kalman equalizer and can be used as EQA. The junction points of KEQA to outer circuits are, as shown in FIGS. 1 and 2, P₅, P₆ and P₇ which are input point of automatic equalizer, input point of reference signal and output point of automatic equalizer respectively. P₄ is input point of sampler. S₁ is sampler as is shown in FIGS. 1 and 2. In FIG. 5, single lines denote single leads and double lines denote multi leads. TFA and TGMA denote real number transmission function and real number tap gain memory respectively. Both ADA₁ and ADA₂ are real number adders and KCA is real number Kalman controller. TGMA and ADA₂ as a whole, is called real number resister RA.

The operation of the system in FIG. 5 is explained now. Real number equivalent base band transmission line ETRLA output appearing at P₄ is sampled at sampler S₁ at the frequency of every T second. The output signal of S₁ enters the real number transmission function TFA. TFA obtains information from real number tap gain memory TGMA and constitutes transmission function of this real number Kalman equalizer KEQA with the aid of TGMA and generates, at the output point of automatic equalizer P₇, convolution result of equalizer input signal and transmission function of EQA stored at TGMA. The difference of equalizer output and reference signal appearing at P₆ is obtained at the adder ADA₁ according to the sign shown in the figure. ADA₁ output and equalizer input appearing at P₅ enter into real number Kalman controller KCA. KCA calculates variation value of real number tap gain memory TGMA storage and adds the value emerged from KCA to the former storage of TGMA with the aid of adder ADA₂ providing new storage of TGMA. This operation is repeated every time when signal appears at P₅ from S₁. The real number Kalman equalizer KEQA operates to make the output of ADA₁ smaller in the sense of power. The above is rather simple sketch of real number Kalman equalizer KEQA, to which more detailed explanation will follow.

FIG. 6 is more minute expression of real number Kalman equalizer KEQA with re-division, into more small components, of the expression in FIG. 5. In FIG. 6, TDLA₁ and TDLA₂ are real number tapped delay line of length M. M is a prescribed integer and equalization becomes more precise as M becomes larger. MA₁ to MA₈ are real number multipliers. ADA₁ to ADA₅ are real number adders. SUMA₁ and SUMA₂ are real number concentrators. IA, SIA, U and PA are real number inverse value generator, real number sign inverter, unit matrix generator and real number P resister, respectively. S₁ is sampler which, as is in FIG. 4, samples its input signal every T second. KEQA, TFA, TGMA, RA and KCA are as shown before real number Kalman equalizer, real number transmission function, real number tap gain memory, real number resister and real number Kalman controller respectively. They are the same components of the ones of the same notation in FIG. 4. Single lines and double lines denote single leads and multi leads respectively, situation being the same with that in FIG. 4. P₅ to P₇ denote the same point as these in FIG. 4 with the same notations. Operation of real number Kalman equalizer KEQA shown in FIG. 6 will be shown below. First, real number transmission function TFA and real number resister RA will be dealt with. Real number tapped delay line TDLA₁ will generate input vector x_(k) shown in following Eq. (1), where input signal at input point of automatic equalizer P₅ and at the time k (k is time variable of integer value) is denoted by g_(k). ##EQU1## Real number tap gain memory TGMA memorizes tap vector of dimension M, h_(k) which is the state of real number Kalman equalizer KEQA. Real number multiplier multiplies corresponding elements of x_(k) and h_(k). Real number concentrator sums up the MA₁ output. MA₁ and SUMA₁, working in cooperation, operate the algebra of

    y.sub.k =x.sub.k.sup.T h.sub.k                             (2)

and gives scalar number y_(k). T shows matrix transpose in this invention. Consequently the output signal of real number Kalman equalizer is y_(k) given in Eq. (2). The action of real number adder ADA₂ will be shown at the passage concerning real number Kalman controller KCA. Operation of real number Kalman controller KCA will be shown now. Real number tapped delay line TDLA₂ have the same construction as that of TDLA₁ and its output is naturally input vector x_(k). Real number P resister PA stores M×M matrix P_(k) which is given afterwards. Real number multiplier MA₂ multiplies P_(k) which is output of real number P resister PA with input vector x_(k) generating M dimension vector P_(k) x_(k). Real number multiplier MA₃ multiplies elements of output vector of MA₂ with corresponding elements of input vector x_(k) which is output of real number tapped delay line TDLA₂. Real number concentrator SUMA₂ sums up multi-leaded output of MA₃. MA₂ and SUMA.sub. 2 working together generate scalar output of x_(k) ^(T) P_(k) x_(k). Point R₂ is input point of parameter r₂ which can be fixed value in the usual applications. This point is supposed to be connected to some appropriate circuit which generates the value r₂. Real number adder ADA₃ adds the value r₂ and output of real number concentrator SUMA₂. Real number inverse value generator IA generates inverse value of ADA₃ output. Real number multiplier MA₄ multiplies M dimensional vector P_(k) x_(k) which is output of MA₂ with scalar output of IA. When output value of MA₂ is called as K_(k), this value can be given as; ##EQU2## as a result of formerly mentioned situation. Real number multiplier MA₅ multiplies M dimensional vector K_(k) and error output of the equalizer e_(k) given at the output point of ADA₁ giving rise to the result Δh_(k) which is M dimensional vector given by;

    Δh.sub.k =K.sub.k e.sub.k                            (4)

When reference signal is denoted by y_(k), ADA₁ output e_(k) will be of course given by;

    e.sub.k =y.sub.k -y.sub.k                                  (5)

Real number adder ADA₂ conducts the operation of adding MA₅ output Δh_(k) to tap gain vector h_(k) stored in real number tap gain memory TGMS generating storage of TGMA at the next time slot which is k+1. Namely, the operation

    h.sub.k+1 =h.sub.k +Δh.sub.k                         (6)

is done. The configuration in FIG. 6, on the other hand, has the other adjustment portion. Operation concerning this portion will be explained hereafter. Output P_(k) of real number P resister PA and output x_(k) of real number tapped delay line are multiplied at real number multiplier MA₆ generating M dimensional vector x_(k) ^(T) P_(k). Multiplication between MA₆ output and MA₄ output which is K_(k) and is given in Eq. (3) is conducted at real number multiplier MA₇ giving rise to M×M matrix K_(k) x_(k) ^(T) P_(k). MA₇ output is inverted in sign by real number sign inverter SIA and then added to real number P resister PA output P_(k) at real number adder ADA₄ giving rise to P_(k) -K_(k) x_(k) ^(T) P_(k). A scalar r₁ introduced at point R₁ as a parameter and unit matrix I generated at unit matrix generator U are multiplied at real number multiplier MA₈ resulting in r₁ I, which is then added to ADA₄ output at the adder ADA₅. ADA₅ output is then P_(k) -K_(k) x_(k) ^(T) P_(k) +R₁ I. Point R₁ is fed with fixed value r₁ and supposed to be connected to proper circuit generating value of r₁ as in the case of point R₂. ADA₅ output is now future content of real number P resister PA and is P_(k+1) given by;

    P.sub.k+1 =P.sub.k -K.sub.k x.sub.k.sup.T P.sub.k +r.sub.1 I=(I-K.sub.k x.sub.k.sup.T)P.sub.k +r.sub.1 I                          (7)

Real number Kalman equalizer shown is FIG. 6 reads data every T second using sampler S₁ and updates content of real number P resister PA and content of real number tap gain memory TGMA h_(k) with the aid of real number Kalman controller KCA and, as a result, brings h_(k) which is content of TGMA nearer to the inverse circuit of real number equivalent base band transmission line ETRLA and makes error signal e_(k) smaller. Parameters R₁ and r₂ are used to make the operation of real number Kalman equalizer KEQA optimum. When time variation of ETRLA is large, R₁ is chosen to be value other than 0. Value of r₂ must be, on the other hand, chosen to be near to noise power generated in ETRLA for obtaining fast convergence of KEQA to the inverse circuit of ETRLA. The system in FIG. 6 will reach to convergence after M time slots when there are no variation of ETRLA and no noise. Of course, M is number of taps in real number tapped delay lines TDLA₁ and TDLA₂. This is prominent advantage of Kalman equalizer as a whole. This system, nevertheless, needs M² multiplications for one time slot in each of real number multipliers MA₆ and MA₇ resulting in large amount of operational labour. This will be the main reason why Kalman equalizer is not put into actual use in spite of its excellent fast convergence characteristics.

Complex number automatic equalizer like EQQ in FIGS. 3 and 4 which deals with impulse response of complex number accompanying to QAM modulation will be treated hereafter. The equalizer will be divided into fairly large sections as shown in FIG. 7. KEQQ is complex number Kalman equalizer which is used as complex number automatic equalizer EQQ in FIGS. 3 and 4. KEQQ has junction points Q₅, Q₆ and Q₇ to outer circuit which are the same points as corresponding ones in FIGS. 3 and 4 and which denote input point of automatic equalizer, input point of reference signal and output point of automatic equalizer respectively. Q₄ is input point of sampler. S₂ is sampler which is the same part as one in FIGS. 3 and 4 with the same notation. As is in FIG. 5, single lines indicate single lead and double lines multiple lines in FIG. 7. TFQ is complex number transmission function and TGMQ is complex number tap gain memory. ADQ₁ and ADQ₂ denote complex number adders. KCQ represents complex number Kalman controller. Both TGMQ and ADQ₂ as a whole are called complex number resister RQ. The operation of the system will be discussed now. Transmission line output appears at Q₄ and sampled at the interval of T second by the sampler S₂. The output of S₂ enters in the complex number transmission function TFQ. TFQ obtains information from complex number tap gain memory TGMQ and provides transmission function of this automatic equalizer with the aid of TGMQ and generates at output point of automatic equalizer Q₇ convolved value of input signal and transmission function of automatic equalizer stored in TGMQ. Complex number adder produces difference between output of the complex number Kalman equalizer KEQQ and reference signal introduced at Q₆ according to the sign notation shown in the figure. ADQ₁ output and equalizer input appearing at Q₅ are introduced into complex number Kalman controller KCQ. Using these two inputs, KCQ calculates change of the content of complex tap gain memory TGMQ and adds this value to the former content of TGMQ and as a result provides new storage of TGMQ. These operations are repeated every time when signal appears at Q₅ through S₂ and KEQQ operates to make the ADQ₁ output power smaller.

These are summary of operation of complex number Kalman equalizer KEQQ. Here more detailed description will be made using FIG. 8. In FIG. 8, points X₁ to X₁₈ are defined. These points are used in figures beginning with FIG. 9 to show the relation with FIG. 8. Superficially FIG. 8 differs from FIG. 6 which represents composition of real number type in that complex conjugate generator C is added between points X₁ and X₂. Following is the description of FIG. 8.

All situation described in FIGS. 5 and 6 remains the same with KEQQ in FIG. 8. Nevertheless, in the system shown in FIGS. 7 and 8, the values dealt with are complex ones and all multipliers and adders are altered to process complex numbers. Though there is little difference between construction of FIGS. 5 and 6 and that of FIGS. 7 and 8, substantially a large change has took place. The complex number Kalman equalizer KEQQ shown in FIGS. 7 and 8 is not only used in QAM modulation system but also used as indispensable component of bandwidth division equalizers which is component of this invention. C in FIG. 8 is complex conjugate generator. KEQQ is of course complex number Kalman equalizer which can be used as KQQ in FIGS. 2 and 3. Summarized operation of the system of FIG. 8 according to sections under fairly large division, which are TFQ, RQ, ADQ₂ and KCQ, has been explained. These parts are expressed by notations used in corresponding parts in FIG. 6 with the change at last letters from A to Q showing that complex numbers are dealt with. Detailed illustration of each composing part in FIG. 8 will be done next. TDLQ₁ and TDLQ₂ in FIG. 8 are complex number tapped delay lines tap number of which are M (M is a prescribed integer). MQ.sub. 1 to MQ₈ are complex number multipliers. ADQ₁ through ADQ₅ are complex number adders. SUMQ₁ and SUMQ₂ are complex number concentrators. IQ, SIQ, U and PQ are complex number inverse value generator, complex number sign inverter, unit matrix generator and complex number P resister respectively. S₂ is also sampler sampling its input signal at the frequency of T second. As is in FIG. 7 single lines designate single leads and double lines multiple leads.

Following is the operation of complex number Kalman equalizer shown in FIG. 8. As the operations of component parts of this KEQQ equalizer are the same as these of component parts of the same names but the last letters being changed from Q to A in real number Kalman equalizer KEQA, explanation of operation is considerably omitted. Nevertheless, as the consequence of introduction of complex conjugate generator C and admission of complex values, Eq. (3) has been changed to;

    K.sub.k =P.sub.k x.sub.k /(x.sub.k.sup.T P.sub.k x.sub.k +r.sub.2) (3)'

The other equations can be used with no alternations except that characters in these equations now represent complex numbers. Namely, Eqs. (1), (2), (4), (5), (6) and (7) remain the same as equations dealing with complex numbers and Eq. (3) is replaced by Eq. (3)'. Substituting Eq. (3)' into Eq. (7), we get

    P.sub.k =P.sub.k -P.sub.k x.sub.k.sup.T x.sub.k P.sub.k /(x.sub.k.sup.T P.sub.k x.sub.k +r.sub.2)+R.sub.k I                       (8)

The terms x_(k) ^(T) x_(k) and x_(k) ^(T) P_(k) x_(k) in the above equation provide power of complex number time function and are apparently of positive real numbers showing that correct function of the system is assured. Complex conjugate generator C is indispensable for the adoption of complex number time functions.

The more detailed composition of the system shown in FIG. 8 is illustrated in the succession of figures beginning with FIG. 9. Here, the case where M which is tap number of the tapped delay lines TDLQ₁ and TDLQ₂ is restricted to 3 is dealt with from reason of simplicity of explanation. Nevertheless M can be chosen arbitrary and so composition shown here does not limit the arbitrariness of M. FIG. 9 is detailed figure of compound type expression of complex number tapped delay lines TDLQ₁ and TDLQ₂. Points X₃ and X₁ are composed of three points respectively and component points of X₃ for example are called as X₃₁, X₃₂ and X₃₃ in the order of less delay as shown in FIG. 9. The term "compound type" means the description method in which real part and imaginary part which are composing elements of complex number are expressed lumped together. On the other hand the term "separate type" denotes the method where these elements are dealt with separately. On FIG. 9 a single lead carrying complex number is written in a real line and so each real line is consisted of two lines; namely, a real number line and an imaginary number line. Of course expression in FIG. 8 is of compound type. DLQ's in FIG. 9 represent unit delay element of T second for complex number. FIG. 10 handles the same part as in FIG. 9 with separate type expression. Here, dotted lines represent leads for real number and chain lines for imaginary number. This expression method is applied similarly to other figures. In FIG. 10, points X₃₁ and X₁₁, for example, are re-divided into two points respectively. The component shown as T is unit delay element of T second. Both of two illustration methods, compound type and separate type, will be used here to provide maximum clarity according to situations. TDLQ₁ and TDLQ₂, as easily recognized from FIG. 10, can be realized by shift resisters operating on T second clock. FIG. 11 shows the composition of complex number multiplier MQ₁ in compound type expression. At the detailed description of the component of the system KEQQ shown in FIG. 8, constituting points of X_(i) (i is an integer) will be expressed either with X_(ij) or with X_(ijk) (i, j and k are integers and j and k take values between 1 to M where M is 3) sub-subscripts being used. The mutual positional relations among these points are kept the same over these figures. MQ_(1i) 's (i= 1, 2, 3) which comprise MQ₁ can be expressed by unit complex number multipliers. MQ₁₁, for example, can be expressed as shown in FIG. 12. M₁₁₁ through M₁₁₄, ADA₁₁ and ADA₁₂ and SIM₁₁ in FIG. 12 are unit multipliers for real number, ordinary unit adders for real number and ordinary sign inverter for real number respectively. These parts can be realized through the use of the hardwares similar to the component of widely used electronic calculators. FIG. 13 is description of complex number concentrator SUMQ₁ by separate type expression. ADAS₁₁ and ADAS₁₂ are ordinary adders for real number. FIG. 14 shows section comprised with complex number adder ADQ₂ and complex number tap gain memory TGMQ or the section of complex number resister RQ with separate type expression. ADA₂₁₁ through ADA₂₃₂ and TGM₁₁ through TGM₃₂ are unit adders and RAM (Random Access Memory) for one character respectively. FIG. 15 represents composition of complex conjugate generator C in separate type expression. SIQ₁ through SIQ₃ are unit sign inverters. For the purpose of providing understanding of processing for complex number and showing the situations in complex algebra in actual operation, the operations using real number decomposition have been shown. FIG. 10 shows delay operation using decomposition into real number as compared to complex number operation shown in FIG. 9 and FIG. 12 gives the multiplication under the way of real number decomposition contrasting complex number operation in FIG. 11 and FIG. 13 shows the method of complex number addition using real number decomposition. Furthermore FIG. 14 shows denoting method of complex number and FIG. 15 depicts generation of complex conjugate value by the real number decomposition. As the operation methods for complex number seems to be sufficiently described, the detailed illustration of the composing parts in FIG. 8 will henceforth be given under compound type expression for the purpose of clarity of description. FIG. 16 is figure for complex number P resister PQ. P₁₁ through P₃₃ are composed by RAM's and store corresponding complex elements in matrix P_(k). FIG. 17 is minute description of complex number multiplier MQ₂ in compound type expression. MQ₂ is, as formerly seen, the part where multiplication between 3×3 matrix P_(k) and vector x_(k) of dimension 3 is conducted generating vector P_(k) k_(k) of dimension 3. MQ₂₁₁ through MQ₂₃₃ here are complex multipliers shown in FIG. 12. ADM₂₁, ADM₂₂ and ADM₂₃ are complex number adders. FIG. 18 depicts composition concerning MQ₃ and SUM₂. MQ₃₁, MQ₃₂ and MQ₃₃ are complex number multipliers and SUMQ₂ is comprised of only one adder SUMQ₂. As complex adder can be realized by unit complex number adder and invertor which provides inversion of positive real value can be materialized by the hardware similar to the ones used in electronic calculator conducting inverse operation, further explanation will be curtailed here. Complex number multiplier MQ₄ is expressed as shown in FIG. 19. MQ₄₁ through MQ₄₃ here are unit complex number multipliers. MQ₅ takes the same configuration as that of MQ₄, where X₁₄ corresponds to X₁₂ and X₁₀ to X₁₃ and X₁₃ to X₇. FIG. 20 is detailed expression of complex number multiplier MQ₆. MQ₆₁₁ through MQ₆₃₃ and ADM₆₁ through ADM₆₃ are complex number multipliers and complex number adders respectively. FIG. 21 is the figure of complex number multiplier MQ₇. MQ₇₁₁ through MQ₇₃₃ are complex number multipliers. The detailed figure for complex number sign inverter SIQ is omitted because of self evidence of the operation. Complex number adders ADQ₄ and ADQ₅ can be composed of by nine pieces of unit complex number adders respectively and explanation is skipped for the reason of obviousness. Unit matrix generator U can be constructed as shown in FIG. 22. U₁₁ through U₃₃ which are one character memories can be comprised with ROM(Read Only Memory)'s with U₁₁, U₂₂ and U₃₃ producing 1's and other parts producing 0's. Complex number multiplier MQ₈ takes the shape shown in FIG. 23. MQ₈₁₁, MQ₈₂₂ and MQ₈₃₃ can be composed of complex number multipliers. These are detailed explanations for the composing parts of the system in FIG. 8 which is complex number Kalman equalizer KEQQ and it is easily understandable that all the components can be realized by already known hardwares. Real number Kalman equalizer KEQA depicted in FIGS. 5 and 6 and used for AM modulation system can handle only real number symbol and, as a result of the fact that coefficients included in the KEQA are of real number value, can operate only when the impulse response of the equivalent base band transmission line has real impulse response. Complex number Kalman equalizer KEQQ for QAM modulation, on the other hand, can operate even when both of or either of above mentioned two kinds of values are not of real numbers and can provide inverse circuit of complex number equivalent transmission line ETRLQ.

Next, widely known theories concerning fast Kalman equalizer which is the other candidate for comprising real number equalizer will be shown. This type of equalizer serves as a basis of complex number fast Kalman equalizer which composes a part of this invention. The result of dividing the equalizer which is called as FKEQA into fairly large sections is shown in FIG. 24. This can be used as real number automatic equalizer EQA shown in FIGS. 1 and 2. FKCA in FIG. 24 indicates real number fast Kalman controller. DLA₁ and DLA₂ are real number delay elements with delay of T second. The other composing parts are the same as those in real number Kalman equalizer KEQA shown in FIG. 5, namely TFA is real number transmission function and TGMA is real number tap gain memory and ADA₁ and ADA₂ are real number adders. As is in FIG. 5, the part built by TGMA and ADA₂ is called real number resister RA. P₄, P₅, P₆ and P₇ are input point of sampler, input point of automatic equalizer, input point of reference signal and output point of automatic equalizer respectively and S₁ is sampler. Almost all situation remains same compared with that in FIG. 5 or real number Kalman equalizer KEQA. The difference between real number fast Kalman equalizer FKEQA in FIG. 24 and real number Kalman equalizer shown in FIG. 5 will be discussed now. FKEQA and KEQA have real number fast Kalman controller FKCA and real number Kalman controller KCA as controlling parts respectively and controllings are conducted through mutually very different theories of fast Kalman filter and Kalman filter respectively. Nevertheless, RA and TFA operate similarly. The other difference lies in that FKEQA has additional parts of real number delay elements DLA₁ and DLA₂. The real number delay element DLA₁ is introduced because of the fact that real number fast Kalman controller FKCA need, for its operation, input signal to real number transmission function TFA T second before the entering of the signal in TFA and DLA₂ is introduced to compensate the time difference between signal at equalizer output and reference signal appearing at point P₆. The difference is caused by introduction of DLA₁. The more detailed illustration of FKEQA shown in FIG. 24 is given in FIG. 25. Explanation according to FIG. 25 is now given. The portions dealt with in FIG. 24 remain the same in the figure. Multi leads which are expressed by double lines are henceforth, beginning with FIG. 24, expressed by single lines for the sake of avoidance of complexity. Moderately minute sectioning is applied to the real number fast Kalman controller FKCA dividing it into parts of PTAA, PTSA, PTDA and PTkA which are, respectively, real number A resister controller, real number S resister controller, real number D resister controller and real number k resister controller. TDLA₁ is, as shown before, real number tapped delay line having M taps and TDLA₂, on the other hand, tapped delay line of M-1 taps. ARA, SRA, DRA and kRA are real number A resister, real number S resister, real number D resister and real number k resister respectively. MA₁ and MA₁₁ through MA₂₂ are real number multipliers. SUMA₁ is, as before, is real number concentrator. ADA₁, ADA₂ and ADA₁₁ through ADA₁₉ are real number adders. INA₁ and INA₂ are real number inverse value generators and I is unit number generator. DIA₁ and DIA₂ are real number dimension increaser. DLA₃ and DLA₄ are, like DLA₁ and DLA₂, real number delay elements.

Following is the description of operation of the real number fast Kalman equalizer FKEQA. First the operation of real number transmission function TFA and real number resister RA is dealt with. Real number tapped delay line TDLA₁ provides, under the influence of DLA₁, input vector x_(k) defined by following Eq. (9), where g_(k) is given as input signal appearing at the point P₅ at the k'th time slot (k is of course time variable with integer value). ##EQU3## Real number tap gain memory TGMA stores tap gain vector h_(k) of dimension M which is the state of this real number fast Kalman equalizer FKEQA. Real number multiplier MA₁ does multiplication between corresponding elements of x_(k) and h_(k) whereas real number concentrator SUMA₁ sums up the output signals from MA₁. As a consequence, MA₁ and SUMA₁, as a whole, conduct the operation of following Eq. (10) giving rise to scalar y_(k).

    y.sub.k =x.sub.k.sup.T h.sub.k                             (10)

The output signal of real number transmission function TFA is y_(k) given in Eq. (10). Error signal e_(k) of the real number fast Kalman equalizer FKEQA is given as difference between y_(k) and y_(k). The difference appears at the output point of real number adder ADA₁ given by following Eq. (11).

    e.sub.k =y.sub.k -y.sub.k                                  (11)

y_(k) is reference signal entering from point P₆. As mentioned before, real number adders to which sign symbols + and - are attached function as subtractors which do subtraction according to the sign attached. Updating of h_(k) is conducted at real number adder ADA₂ making use of vector k_(k) of dimension M which is output signal of real number k resister kRA and scalar value e_(k) shown above. That is to say, incremental component of h_(k) which is Δh_(k) given in the following Eq. (12) and obtained at the output of MA₂₂ is used to obtain h_(k+1) which is the value of h_(k) at the next time slot and is given in following Eq. (13) and which appears as the output of ADA₂.

    Δh.sub.k =k.sub.k e.sub.k                            (12)

    h.sub.k+1 =h.sub.k +Δh.sub.k                         (13)

The contents in real number k resister kRA which is k_(k) is also updated at every time slot. These types of updatings occur at many places in the real number fast Kalman controller FKCA. Following is the explanation of these updatings. First, description of k_(k) using its component elements is given for the convenience of explanation in following Eq. (14). ##EQU4## From now on the operation of FKCA will be depicted according to the order of processing. This part admits input signal g_(k) appearing at input point of automatic equalizer P₅ as its input and provides incremental information Δh_(k) of the stored value at the real number tap gain memory TGMA as its output. The signal at the input point of real number delay element DLA₁ is that at P₅ and is g_(k). Output signal of real number tapped delay line TDLA₂ is M-1 dimensional vector and is expressed by: ##EQU5## Output of real number delay unit DLA₄ is apparently g_(k-M). The value g_(k-1) which is earlier by one time slot compared to presently incoming signal g_(k) is needed at DLA₁ in FKCA. As formerly explained DLA₁ and DLA₂ are used for the purpose of time adjustment. g_(k) and u_(k) are introduced into real number dimension increaser DIA₁ and similarly u_(k) and g_(k-1) into dimension increaser DIA₂. DIA₁ provides vector c_(k+1) having dimension larger by amount of one than that of u_(k). Output vector of real number tapped delay line TDLA₁ which is x_(k) is expressed by Eq. (9). x_(k+1) is the state of x_(k) later by one time slot which is of course expressed as: ##EQU6## Output of real number dimension increaser DIA₂ is x_(k) given in Eq. (9). Using these g_(k), x_(k+1), x_(k), g_(k-M) and e_(k) which is output of ADA₁, PTAA, PTSA, PTDA and PTkA, which are real number A resister controller, real number S resister controller, real number D resister controller and real number k resister controller respectively, function. These parts conduct adjustment of the contents of real number A resister ARA, real number S resister SRA, real number D resister DRA and real number k resister kRA. Above mentioned controllers act their operation at the same order of mentioning them or at the top to bottom order of the configuration in FIG. 25. First, action of real number A resister controller PTAA will be shown. Input signals to the part are g_(k) which is input signal to real number delay line DLA₃, x_(k) which is output signal of real number dimension increaser DIA₂ and formerly mentioned k_(k) which is output signal of real number k resister kRA. Real number A resister ARA, at this instance, stores A_(k-1) which is value for former time slot. A_(k-1) is vector with M dimension and given by: ##EQU7## By the use of real number multiplier MA₁₁ and real number adder ADA₁₁ and also signals of A_(k-1), x_(k) and g_(k), scalar value q_(k) expressed in following Eq. (18) is obtained.

    q.sub.k =g.sub.k +A.sub.k-1 x.sub.k                        (18)

By means of real number multiplier MA₁₂, real number adder ADA₁₂ and using, as input signals, k_(k) and q_(k), q_(k) being output of ADA₁₁, the content of real number A resister ARA which is of course A_(k) is calculated by:

    A.sub.k =A.sub.k-1 -k.sub.k q.sub.k.sup.T                  (18)'

This value is introduced into ARA completing the update. After this instant ARA will give the value of A_(k). This is the action of real number A resister PTAA.

Real number S resister controller PTSA is analyzed next. The input signals for this part are q_(k) which is output of real number adder ADA₁₁, A_(k) which is output of real number A resister ARA, g_(k) which is input signal to real number delay element DLA₃ and x_(k) which is output of real number dimension increaser DIA₂. Employing real number multiplier MA₁₄ and real number adder ADA₁₃, the value v_(k), a scalar, is produced at the output of ADA₁₃. v_(k) is given by:

    v.sub.k =g.sub.k +A.sub.k.sup.T x.sub.k                    (19)

At real number multiplier MA₁₃ and real number adder ADA₁₄, S_(k) is calculated and content of S resister SRA is updated from S_(k-1) to S_(k), where S_(k) is storage of SRA for next time slot. S_(k) is given by:

    S.sub.k =S.sub.k-1 +v.sub.k q.sub.k                        (20)

The operation of real number S resister controller has been thus explained.

Next, operation of real number D resister controller will be shown. The input signals for this part are v_(k) which is output of real number adder ADA₁₃, S_(k) which is output of real number S resister SRA, A_(k) which is output of real number A resister ARA, k_(k) which is output of real number k resister kRA, g_(k-M) which is output of real number delay element DLA₄ and x_(k+1) which is output of real number dimension increaser DIA₁. Apart from these signals, output of unit number generator I must be used. Utilizing real number inverse value generator INA₁ which generates inverse value of its input and real number multiplier MA₅, scalar value S_(k) ⁻¹ v_(k) is generated as output of MA₅. S_(k) ⁻¹ is inverse value of S_(k). Real number multiplier MA₁₆ will do the multiplication of A_(k) and S_(k) ⁻¹ providing A_(k) S_(k) ⁻¹ v_(k) as its output. This value is of M dimension from the fact that A_(k) is a vector of M dimension. At real number adder ADA₁₅, the sum of A_(k) S_(k) ⁻¹ v_(k) and k_(k), where k_(k) is output of real number k resister kRA, is calculated. The output of adder ADA₁₅ is consequently k_(k) +A_(k) S_(k) ⁻¹ v_(k) which is vector of dimension M. Real number dimension increaser DIA₃ provides vector of dimension M+1 which is composed by arranging vector k_(k) +A_(k) S_(k) ⁻¹ v_(k) of dimension M under the scalar S_(k) ⁻¹ v_(k). The situation can be written as shown in following Eq. (21). ##EQU8## f_(k) in Eq. (21) will be produced at the output of DIA₃. At real number dimension decreaser DDA₁, f_(k) is divided into two components one of which is m_(k) of M dimension and the other of which is scalar w_(k). These are written as: ##EQU9## At this instant real number D resister DRA stores M dimensional vector D_(k-1) which is value for former time slot. D_(k-1), X_(k+1) which is output of real number dimension increaser DIA₁ and g_(k-M) which is output of real number delay element DLA₄ will be used at real number multiplier MA₁₇ and real number adder ADA₁₆ giving rise to a scalar r_(k) given in following Eq. (24).

    r.sub.k =g.sub.k-M +D.sub.k-1.sup.T x.sub.k+1              (24)

From D_(k-1) which is output of DRA, m_(k) which is output of DDA₁ and r_(k) which is output of ADA₁₆, D_(k-1) -m_(k) r_(k) is obtained as an output of ADA₁₈ through the operation of real number multiplier MA₂₀ and real number adder ADA₁₈. Furthermore, based on unit value which is output of unit value generator I, r_(k) which is output of ADA₁₆ and w_(k) which is output of DDA₁, the value [1-r_(k) w_(k) ]⁻¹ is obtained as output of real number inverse value generator INA₂ through the effect of real number multiplier MA₁₈, real number adder ADA₁₇ and INA₂. INA₂ provides, like INA₁, the inverse value of its input at its output. The result of multiplication of [D_(k-1) -m_(k) r_(k) ] and [1-r_(k) w_(k) ]⁻¹ is obtained at the output of real number multiplier MA₁₉. The result is introduced into real number D resister DRA as the new value of DRA which is D_(k) updating DRA. This process is expressed by:

    D.sub.k =[D.sub.k-1 -m.sub.k r.sub.k.sup.T ][1-w.sub.k r.sub.k ].sup.-1 (25)

Following is operation of real number k resister controller PTkA. The input signals for this component are D_(k) which is output signal of real number D resister DRA, w_(k) which is output signal of real number dimension decreaser DDA₁ and m_(k) which also comes from DDA₁. As a result of operations in real number multiplier MA₂₁ and real number adder ADA₁₉, the operation given in following Eq. (26) is done producing k_(k+1) which is storage of real number k resister for next time slot.

    k.sub.k+1 =m.sub.k -D.sub.k w.sub.k                        (26)

k_(k) given in Eq. (14) is M dimensional vector. Real number k resister kRA is now updated and the system of real number fast Kalman equalizer FKEQA waits for the input g_(k+1) of the next time slot.

Now the illustration of operation for the real number fast Kalman equalizer FKEQA which is widely known is completed. Succeeding widely known technic of real number fast Kalman equalizer mentioned above, the technic of complex number fast Kalman equalizer which consists considerable part of this invention will now be explained. This technic can be used to equalize complex number equivalent base band transmission line ETRLQ shown in FIGS. 3 and 4 and constitutes the basis of bandwidth division fast Kalman equalizer shown afterward. The figure for complex number fast Kalman equalizer FKEQQ with sectioning into fairly large sections is FIG. 26. The composition is very similar in its shape to the real number fast Kalman equalizer FKEQA given in FIG. 24.

The complex number fast Kalman equalizer can be utilized, like complex number Kalman equalizer KEQQ formerly explained, as complex number equalizer EQQ shown in FIGS. 3 and 4 and can equalize complex number equivalent base band transmission line ETRLQ, whereas real number fast Kalman equalizer FKEQA is restricted in its use to the action of real number equalizer EQA shown in FIGS. 1 and 2. At FIG. 26 of FKEQQ, FKCQ is complex number fast Kalman controller. DLQ₁ and DLQ₂ are complex number delay elements with T second of delay. Other composing parts remain the same as those in complex number Kalman equalizer KEQQ shown in FIG. 7. Henceforth TFQ is complex number transmission function and TGMQ is complex number tap gain memory, and finally ADQ₁ and ADQ₂ are complex number adders. As is in FIG. 7, GMQ and ADQ₂ lumped in one component is called complex number resister RQ. Q₄, Q₅, Q₆ and Q₇ are input point of sampler, input point of automatic equalizer, input point of reference signal and output point of automatic equalizer respectively and S₂ is sampler. The situation that this equalizer can be utilized as complex number equalizer EQQ shown in FIGS. 3 and 4 is unchanged as is situation of complex number Kalman equalizer KEQQ. The difference between complex number fast Kalman equalizer FKEQQ shown in FIG. 26 and complex number Kalman equalizer KEQQ shown in FIG. 7 lies in that the former operates under the theory of fast Kalman filter making use of complex number fast Kalman controller FKCQ whereas the latter under theory of Kalman filter making use of complex number Kalman controller KCQ. The difference is the same as that between real number fast Kalman equalizer FKEQA and real number Kalman equalizer KEQA. On the other hand, the difference between real number fast Kalman equalizer FKEQA and complex number fast Kalman equalizer FKEQQ explained here, is that the latter uses theory of fast Kalman filter extended to the region of complex number. This expansion is given originally at this invention. The role of complex number delay elements DLQ₁ and DLQ₂ is the same as that of real number delay elements DLA₁ and DLA₂ in the case of real number fast Kalman equalizer FKEQA in FIG. 24 and henceforth explanation will not be repeated. The more detailed configuration of FIG. 26 is given in FIG. 27. Henceforth study of each composing part and its function is made following FIG. 27. In FIG. 27, components already shown in FIG. 26 of course remain the same. Also for complex number Kalman controller FKCQ, middle scale division is applied resulting in the sectioning into PTAQ, PTSQ, PTDQ and PTkQ which are, respectively, complex number A resister controller, complex number S resister controller, complex number D resister controller and complex number k resister controller. TDLQ₁ is complex number tapped delay line of M taps whereas TDLQ₂ is complex number tapped delay line M-1 taps. ARQ, SRQ, DRQ, and kRQ are complex number A resister, complex number S resister, complex number D resister and complex number k resister respectively. MQ₁ and MQ₁₁ through MQ₂₂ are complex number multipliers and SUMQ₁ is complex number concentrator. ADQ₁, ADQ₂ and ADQ₁₁ through ADQ₁₉ are all complex number adders. INQ₁ and INQ₂ are complex number inverse value generators. I is as before unit value generator. DIQ₁ and DIQ₂ are complex number dimension increasers and DDQ₁ is complex number dimension decreaser. DLQ₃ and DLQ₄ are, like DLQ₁ and DLQ₂, complex number delay elements. Each constituting part at FIGS. 26 and 27 corresponds to that in FIGS. 24 and 25 which have the same names except that the last letter is changed from Q to A doing nearly the same operation. One major and vital change is that complex conjugate generator C₂ connected directly after the output port of ADQ₁₃ in FIG. 27 is newly implemented. This component of course provides complex conjugate value of its input at its output. Because of existence of this component the operation of the system at FIG. 27 which is complex number fast Kalman equalizer FKEQQ becomes different from that of FKEQA in FIG. 25. The processing action of complex number fast Kalman equalizer FKEQQ is the same as that of real number fast Kalman equalizer FKEQA except the operation around a variable v_(k). The operations of component parts of KKEQQ are the same as those of corresponding parts of FKEQA except that v_(k) must be changed to v_(k) and the last letters at the abbreviated names of parts must be changed from A to Q and the term "real number" must be changed to "complex number" at their full names. Henceforth further explanation of operation at each component is curtailed here. Nevertheless illustration of action at complex number S resister controller PTSQ after generation of v_(k) at the output of ADQ₁₃ must be added and following description must be given. Owing to the operation at complex number multiplier MQ₄ and complex number adder ADQ₁₃, the scalar value v_(k) is produced at the output point of MDQ₁₃. This operation is expressed by following Eq. (19).

    v.sub.k =g.sub.k +A.sub.k.sup.T x.sub.k                    (19)

This value v_(k) is directly led to the complex conjugate generator C₂ and v_(k) is obtained at its output point. The symbol ˜ used here is of course complex conjugate symbol. Next, through the cooperation of complex number multiplier MQ₁₃ and complex number adder ADQ₁₄, the value S_(k) is calculated as shown in following Eq. (20)' and accordingly the storage of complex number S resister SRQ is changed from S_(k-1) to S_(k) which is storage of this k'th time slot.

    S.sub.k =S.sub.k-1 +v.sub.k q.sub.k                        (20)'

This is operation of complex number S resister controller PTSQ. Equations for FKEQQ remain unchanged in their expressions from these for KEQQ except that Eq. (20) must be replaced by Eq. (20)' and Eq. (21) by following Eq. (21)'. ##EQU10## Equations (9) through (19) and (22) through (26) begin to deal with complex numbers. Superficial difference between systems in FIG. 25 which is FKEQA and that in FIG. 27 which is FKEQQ is limited only to the existence of complex conjugate generator C₂ but latent and large difference occurs in that system in FIG. 27 can handle complex number whereas the system in FIG. 25 can only deal with real number. Complex number fast Kalman equalizer FKEQQ has much expanded its application area to, for example, utilization as complex number EQQ in FIGS. 3 and 4 contrasting to the area for the system in FIG. 25. Detailed expression of the composing parts in FIG. 27 will be shown at the figures beginning with FIG. 28. Here again the case in which tap number for complex number tapped delay line TDLQ₁ is 3 is shown but the configurations in these figures do not restrict the arbitrariness of the number of M. At FIG. 27 points X₅₁ through X₉₀ are defined. These letters are used to show the relations between FIG. 27 and figures beginning with FIG. 28.

FIG. 28 is detailed description of complex number delay elements DLQ₁ through DLQ₄ in expression of compound type. This figure coincides with that of unit complex number delay element as a whole. As explained before, compound type of expression is the way in which real number part and imaginary number part of complex number is dealt with lumped together whereas separate type of expression is the method where these parts are handled separately. Like expression in former examples leads carrying complex number are denoted by single line in the compound cases and leads carrying real number and lead carrying imaginary number are expressed by dotted lines and chain lines respectively in the separate cases. DLQ's in FIG. 28 are complex number delay elements with T second delay. As apparent from FIG. 28 DLQ₁ through DLQ₄ can be realized by shift resisters operating on T second clock. FIG. 29 consists the detailed figure of complex number tapped delay line TDLQ₁ by the expression of compound type. Point X₅₆ is consisted of three points as a result of tap number M being 3. These points are called as X₅₆₁, X₅₆₂ and X₅₆₃ and arranged as shown in the figure by the order of less delay. For DLQ's description has been made concerning complex number delay elements DLQ through DLQ₄. FIG. 30 is minute expression of compound type for the complex number tapped delay line TDLQ₂. X₅₇ is consisted of two points which are called X₅₇₁ and X₅₇₂ by the order of less delay as shown in the figure. FIG. 31 shows the composition of complex number multiplier MQ₁ with compound type of expression. MQ₁₁ through MQ₁₃ in the figure are unit complex number multipliers. It is apparent that each MQ_(1i) 's (i=1, 2, 3) can be expressed by the configuration shown in FIG. 12 when separate type of expression is used. FIG. 32 is minute figure of complex number concentrator SUMQ₁ in compound type of expression. ADQ is complex number adder admitting three input signals. FIG. 33 is compound expression of both of complex number adder ADQ₂ and complex number tap gain memory TGMQ which are lumped together to be complex number resister RQ. Difference of the two types of expression is quite apparent when FIG. 33 is compared with FIG. 14 which uses separate type of expression. ADQ₂₁ through ADQ₂₃ in FIG. 33 are unit complex number adders. TGMQ₁ through TGMQ₃ can be realized by complex number memories of one character. Complex number adder ADQ₁ can be constructed with scalar adders hence explanation for these parts is omitted because of self evidence. FIGS. 34 and 35 are for minute expression in compound type of complex number dimension increaser DIQ₁ and DIQ₂ respectively. These are composed only of wirings between their terminals.

Next, detailed figures for consisting part of complex number A resister controller PTAQ will be treated. FIG. 36 is detailed description, in compound type of expression, of complex number multiplier MQ₁₁. MQ₁₁₁ through MQ₁₁₃ in this figure are, respectively, consisted of single complex number multiplier whereas ADQ is complex number adder with three input terminals. ADQ₁₁ is scalar adder to which explanation is curtailed for the reason of simplicity. FIG. 37 is minute figure for complex number multiplier MQ₁₂ in compound type of expression. MQ₁₂₁ through MQ₁₂₃ are made from single complex multipliers respectively. FIG. 38 is detailed compound type expression of complex number adder ADQ₁₂. This component is consisted of three single complex number adders ADQ₁₂₁ through ADQ₁₂₃. These, as mentioned before, conduct subtraction by addition using sign inversions of signal treated according to + and - signs in the figure. The compound type expression of complex number A resister ARQ in detail is FIG. 39. This part is composed of three single complex number resisters ARQ₁ through ARQ₃. These above are descriptions of complex number A resister controller PTAQ.

Following is depiction of components consisting complex number S resister PTSQ. FIG. 40 is detailed description of complex number multiplier MQ₁₄. This is consisted of unit complex number multipliers MQ₁₄₁ through MQ₁₄₃ and complex number adder ADQ. The description of ADQ₁₃ will be omitted as it is composed only of single complex number adder ADQ₁₃. FIG. 41 is separate type description in detail of complex conjugate generator C₂. This looks like a part of complex conjugate generator C in FIG. 15. SIQ is single sign invertor. Of course dotted lines denote leads carrying real numbers and chain lines leads of imaginary numbers. Explanation for MQ₁₃ and ADQ₁₄ will be omitted because they are consisted only of single complex number multipliers and adders. Also expression of complex number S resister SRQ will be skipped because it is composed of single complex number resisters. Above descriptions are for complex number S resister controller PTSQ and following illustrations are minute expression of components consisting complex number D resister controller PTDQ. FIG. 42 depicts the complex number inverse value generator INQ₁ in detail. M₁₁ through M₁₄ are composed of single real number multipliers. SQ₁₁ is square root generator which generates square root value of its input and can be composed of by the methods similar to the ones in widely used electronic calculators. IN₁₁ is inverse value generator which provides the inverse value of its input and can be composed of circuits similar to the ones in electronic calculators. SIQ is sign invertor having the same configuration as those used in complex conjugate generator in FIG. 41. Complex number inverse value generator INQ₁ as a whole provides the value of (a-jb)/√a² +b² at the point X₇₄ when input value of (a+jb) is applied at the input point of X₇₃. j is of course imaginary number unit. For complex number multiplier MQ₁₅ composed by one signal complex number multiplier, the explanation is omitted. FIG. 43 is detailed description of complex number multiplier MQ₁₆, where MQ₁₆₁ through MQ₁₆₃ are single complex number multipliers. FIG. 44 is minute figure of complex number adder ADQ₁₅ which is built with single adders ADQ₁₅₁ through ADQ₁₅₃. FIGS. 45 and 46 are, respectively, minute figures for complex number dimension increaser DIQ₃ and complex number dimension decreaser DDQ₁. These are composed only of wirings between their terminals. FIG. 47 is detailed description of complex number multiplier MQ₁₇ which is implemented by single complex number multipliers MQ₁₇₁ through MQ₁₇₃ and a complex number adder ADQ. Complex number adder ADQ₁₆, complex number multiplier MQ₁₈ and complex number adder ADQ₁₇ are composed of single adders or multipliers and therefore description of them will be omitted. Unit value generator generating the value of 1 can be made with a ROM (Read Only Memory) storing the value of 1. Explanation for complex number inverse value generator INQ₂ is skipped because of the fact that INQ₂ is identical with INQ₁ formerly expressed. FIG. 48 is detailed figure for complex number multiplier MQ₁₉ where M₁₉₁ and M₁₉₂ are single complex number multipliers. FIG. 49 is figure for complex number multiplier MQ₂₀ in detail, M₂₀₁ through M₂₀₃ being single complex multipliers. FIG. 50 is complex number adder ADQ₁₈ composed of three single adders ADQ₁₈₁ through ADQ₁₈₃ where each single adder operates subtraction by inverting sign of its input signal according to + and - signs in the figure and conducting addition afterward. FIG. 51 is figure in length of complex number D resister DRQ composed of three single resisters DRQ₁ through DRQ₃. Above is the illustration of complex number D resister controller PTDQ.

Following is the comments on component composing complex number k resister controller PTkQ. FIG. 52 is the figure for complex number multiplier MQ₂₁ which is constituted with three single complex number multiplier MQ₂₁₁ through MQ₂₁₃. FIG. 53 is detailed drawing for complex number adder ADQ₁₉ which is made of three single adders ADQ₁₉₁ through ADQ₁₉₃. FIG. 54 is figure for complex number k resister kRQ. This is built with three single complex resisters kRQ₁ through kRQ₃.

These in all are detailed descriptions for the components in FIG. 27, which all are realized by formerly known hardwares. Contrasting with conventional real number fast Kalman equalizer FKEQA which can only handle real input and which can only generate real impulse response characteristics for its transmission function and which can only operate under the condition that signals entering at the information symbol input point is of real value, the complex number fast Kalman equalizer FKEQQ with the configuration in FIG. 27 can operate equalization even when either or all of above mentioned real value restrictions are not met. In other word, this equalizer can be used as equalizer EQQ for QAM modulation system in FIGS. 3 and 4.

Henceforth bandwidth division equalizer is treated and preparatory explanation in given first. FIG. 55 is a region extracted from FIG. 4 necessary for the illustration here. Point Q₁ ' corresponds to output port of adder SUM₄ to which complex number information signal appears. This point is called input point of complex information value contrasting with input point of information symbol Q₁ in FIG. 4. The situation in FIG. 55 where points Q₄, Q₅, Q₆ and Q₇ are input point of sampler, input point of automatic equalizer, input point of reference signal and output point of automatic equalizer has not changed from that in FIGS. 3 and 4. EQQ, ETRLQ and DL₃ are complex number equalizer, complex number equivalent base band transmission line and complex number delay line for obtaining reference signal respectively. The path of reference signal is now illustrated in real line instead of dotted line in FIG. 4 reflecting the fact that this path is as important as main path. As for complex number equalizer in FIG. 55 either complex Kalman equalizer KEQQ shown in FIGS. 7 and 8 or complex number fast Kalman equalizer KKEQQ in FIG. 26 or 27 can be utilized. Taking advantage of capability of handling complex number, bandwidth division type of equalizers are deduced by modification of above two types of equalizers. Accordingly two types of bandwidth division type of equalizers, namely bandwidth division Kalman equalizer deduced from complex number Kalman equalizer and bandwidth division fast Kalman equalizer from complex number fast Kalman equalizer, can be introduced. The explanation here is made only for the case of the former for the simplicity of explanation. The introduction of the latter is evident from that of former. Other types of complex number equalizer, for example learn and identification type of equalizer used widely, can be arranged into bandwidth division type of equalizers. They will not be, nevertheless, treated in this invention because of their inferior behavior. The complex number automatic equalizer EQQ in FIG. 55 conducts action of equalization over the bandwidth amount of 2π/T (radian/second) situated on the range of -π/T to +π/T (radian/second), in consequence of the fact that the equalizer has tapped delay lines of T second tap spacing which is equal in amount to the symbol sending interval of the data transmission system and that it has sampler S₂ conducting sampling at the interval of T second at its input point. The transmission system shown in FIG. 55 thus conforms well to the nature of information source which generates symbol value at the input point of complex information value Q₁ ' every T second and which has, as a result, a frequency characteristics of its output signal repeating its characteristics between -π/T and +π/T (radian/second) at the frequency of 2π/T (radian/second). The bandwidth division type of equalizer deduced from complex number Kalman equalizer KEQQ will be obtained through the use of the configuration of FIG. 56 which utilizes KEQQ as complex number equalizer EQQ. At the figure, KEQQ is expressed by the block diagram shown in FIG. 7. As is in FIG. 7, KEQQ denotes complex number Kalman equalizer and TFQ is complex number transmission function including tapped delay line of T second tap spacing. RQ and KCQ, again, are complex number resister storing tap weights and complex number Kalman controller giving rise to variation information of tap weights. The points Q₁ ' through Q₇ in FIG. 56 are the same points shown in FIG. 55 with the same notations. Next the configuration of FIG. 57 will be introduced. The difference between this figure and FIG. 56 lies in that there exists band pass filter FILN after the input point of complex information value Q₁ '. FILN has pass band bandwidth of 2π/NT (radian/second) which amounts one N'th of the equalizing frequency bandwidth of 2π/T (radian/second) of the complex number Kalman equalizer KEQQ. N is arbitrary chosen integer. For tapped delay line in the complex number transmission function TFQ in configuration of FIG. 57, or more precisely complex number tapped delay line TDLQ₁ in TFQ illustrated in FIG. 8, necessary tap spacing becomes to be NT (second) instead of original T (second). This is resulted from the fact that equalizer, effective equalization of which is expected only at the bandwidth of 2π/NT (second) must have tapped delay line with tap spacing as small as NT (second). This is proved by following facts. A transversal filter with taps of NT second tap spacing repeats its frequency characteristics at 2π/NT (radian/second) interval. Consequently, if the tap weights of the transversal filter are consisted of complex number, the transversal filter can provide arbitrary characteristics for a bandwidth of 2π/NT (radian/second) at arbitrary position of frequency axis. As a result, transversal filter with complex number tap weights and NT second tap spacing can provide arbitrary transmission characteristics for any band limited signal with bandwidth restricted to 2π/NT (radian/second). For the above mentioned reason, tap spacing at the complex number tapped delay line TDLQ₁ contained in the complex number transmission function TFQ in FIG. 57 needs to be only NT second. These are the proofs. Transversal filter of NT second of tap spacing but with real number tap weight does not have above mentioned property. This is due to the fact that this type of transversal filter can only have frequency characteristics which is symmetrical with the frequency spots of 2πi/NT (radian/second), where i is any integer, and can not provide arbitrary characteristics at arbitrary position of frequency axis. Based on above mentioned considerations, the system shown in FIG. 58 is introduced. Henceforth the operation of this system will be discussed. KEQQN and its composing parts of TFQN, RQN, and KCQN are respectively, a kind of, complex number Kalman equalizer, complex number transmission function, complex number resister and complex number Kalman controller. Constructions of these components are different from corresponding parts in FIG. 57 in that tap spacing involved in complex number transmission function TFQN and complex number Kalman controller KCQN is altered to NT second and accompanying changes such as tap number alternation are done. KEQQN, TFQN, RQN and KCQN are called Kalman equalizer with NP tap spacing, transmission function with NT tap spacing, resister with NT tap spacing and Kalman controller with NT tap spacing respectively. Components in FIG. 58 which are not suffered from name alternation are denoted with the same names and abbreviations as those in FIG. 57. S₂, as before, conducts sampling at T second frequency.

Because transmission function provided from associated operation of transmission with NT tap spacing TFQN and resister with NT tap spacing RQN can give, as formerly mentioned, arbitrary frequency characteristics on the pass band of band pass filter FILN, RQN at the Kalman equalizer with NT tap spacing can be brought into convergence by controlling RQN so that complex number adder ADQ₁ does not produce any output. One thing to be remarked is that state of RQN at the convergence is the same when position and bandwidth of band pass filter FILN on the frequency axis is the same, irrespective of characteristics of FILN on its passband. This is due to the fact that Kalman equalizer with NT tap spacing KEQQN must equalize complex number equivalent base band transmission line ETRLQ over whole pass band of FILN and that goal of convergence for RQN must be unique.

It is evident that boundary conditions for Kalman equalizers with NT tap spacing KEQQN are the same both for FIG. 59 and FIG. 58, as constitution of FIG. 59 is obtained from that of FIG. 58 by moving FILN in FIG. 58, over the junction point, as near as possible to KEQQN with the side effect of increase of number of FILN to 2. The exchange of position between sampler S₂ and band pass filter FILN is possible because the bandwidth treated is limited to 2π/T second and also because whole system is linear one. Q₄ and Q₅ are again input point of sampler and input point of automatic equalizer respectively. The system in FIG. 59 consists the fundamental structure of bandwidth division type of equalizers. The synthesized result of above mentioned technics is bandwidth division Kalman equalizer consisting a major part of this invention.

A case of realization of this invention is given in FIG. 60. This is the figure of bandwidth division Kalman equalizer BDKEQ which operates through division of frequency bandwidth ranging from -π/T (radian/second) to π/T (radian/second) to N sub-bandwidth. Q₁ ', Q₄, Q₅, Q₆ and Q₇ in FIG. 60 are, as before, input point of complex information value, input point of sampler, input point of automatic equalizer, input point of reference signal and output point of automatic equalizer respectively. FILN(0J)'s (J is an integer between 1 to N) are band pass filter with rectangular frequency gain characteristics and no phase characteristics, which are represented with characteristics for FILN(01), FILN(0J) and FILN(0N) at FIGS. 61(a), (b) and (c) respectively. The whole sum of transmission characteristics of these FILN(0J)'s is that of through (direct connection) circuit. KEQQ(01) through KEQQ(0N) are Kalman equalizers with NT tap spacing mentioned before and are the same in their configuration. KEQQ(0J), for example, is called (0J) Kalman equalizer with NT tap spacing according to the number attached. Similarly, FILN(0J) is called as (0J) band pass filter. In FIG. 60, composition of KEQQN(0J) is shown as represented by composition of KEQQN(01). TFQN(01), RQN(01), KCQN(01) and ADQ₁ (01) in FIG. 60 are called, for example, (01) transmission function with NT tap spacing, (01) resister with NT tap spacing, (01) Kalman controller with NT tap spacing, and (01) complex number adder respectively. Junction points KEQQN(01) to outer circuit, for example, is designated in the manner where Q₅ (01), Q₆ (01) and Q₇ (01) are input point of (01) automatic equalizer, input point of (01) reference signal and output point of (01) equalizer respectively. I(01), for example, is the point added for the purpose of explanation and is output point of error signal at KEQQN(01) and called output point of (01) error signal. The configuration of FIG. 60 for bandwidth division Kalman equalizer BDKEQ is consisted of parallel connection of many complex number Kalman equalizers and consequently this configuration can converge because each component equalizer KEQQN(0J) is proved to have ability of convergence. There is no error signal at the output of each I(0J) when convergence is attained at KEQQN(0J). For the purpose of investigating behavior of bandwidth division Kalman equalizer BDKEQ after convergence, FIG. 60 is rewritten into FIG. 62 in which more clear expression of signal path is provided. Taking into account of the fact that convergence is attained and that complex information value appears at Q₁ ' every T second, it will be concluded that output of each I(0J) terminal is 0 and that output signal of FILN(0J) connected to complex number delay line DL₃ is identical to the output signal of TFQN(0J). From self evident fact that two sum totals of signals are mutually identical when each corresponding component consisting these two sum totals is mutually identical, two output signals in FIG. 63 obtained by modification of FIG. 62, namely output from Q₇ and output from point I, are identical. Point I is called error output point of automatic equalizer. SUMR is, like SUMT, concentrator summing N signals. Resulting from the characteristics shown in FIG. 61(d), circuit consisted from FILN(0J)'s connected to complex number delay line DL₃ and SUMR reduces to through (direct connection) circuit. Taking into consideration of the fact that signal at input point of complex information value Q₁ ', with some delay, emerges at point I and that signal at the output point Q₇ of BDKEQ in convergence condition is identical with signal at point I, it will be concluded that signal at point Q₇ is very signal needed as equalizer output and that BDKEQ illustrated in FIG. 60 is necessary equalizer circuit. A way of realizing this invention other than the one shown in FIG. 60 or another example of this invention which makes use of other composing method of bandwidth division Kalman equalizer BDKEQ with other way of bandwidth division, is shown in FIG. 64. Contrasting to the system of FIG. 60 where N band pass filters FILN(01) through FILN(0N) are used, the system in FIG. 64 has 2N band pass filters and accordingly 2N Kalman equalizer with NT tap spacing KEQQ's. For each KEQQN corresponding to each band pass filter FILN(iJ) (i=1, 2; J=1 to N), number is attached, as is in FIG. 60, in the method like KEQQN(iJ) etc. Though the system in FIG. 64 is not different especially from system in FIG. 60 in the sense of configuration, each characteristic of FILN(iJ) (i=1, 2; J=1 to N) is drastically different from that of FILN(0J) (J=1 to N) in FIG. 61 in that characteristics in FIGS. 65 and 66 which are those of FILN(iJ)'s (i=1, 2; J=1 to N) are much gentle in shape compared with that of FILN(0J)'s. In FIGS. 65(a), (b) and (c), frequency gain characteristics of FILN(11), FILN(1J) and FILN(1N) are given. FILN(1J)'s (J=1 to N) are called as band pass filters of first series. Frequency gain characteristics for FILN(21), FILN(2J) and FILN(2N) are shown in FIGS. 66(a), (b) and (c) respectively. These are called band pass filters of second series. These FILN(iJ)'s (i=1, 2; J=1 to N) have no phase characteristics. From the fact that each band pass filter has pass bandwidth of 2π/NT (radian/second), formerly mentioned situation that these filters cooperate with Kalman equalizer with NT tap spacing giving rise to appropriate equalizer operation, remains unaltered. Band pass filters of first and second series compensate each other at their common pass band resulting in flat transmission characteristics as the sum of them. Total sum of 2N transmission characteristics provided by 2N filters FILN(11) through FILN(2N) is reduced again to that of through (direct connection) circuit shown in FIG. 61. Thus, as a consequence of the facts discussed concerning with FIGS. 62 and 63, bandwidth division Kalman equalizer BDKEQ given in FIG. 64 can conduct equalization. Band pass filters FILN(iJ) (i=1, 2; J=1 to N) which are used in the system of FIG. 64 and characteristics of which are given in FIGS. 65 and 66, have more gentle characteristics and, as a result, have a merit of needing less tap number when constructed by transversal filter configuration, compared to the band pass filters FILN(0J)'s (J=1 to N) which are used in the composition of FIG. 60 and whose characteristics are shown in FIG. 61. Necessary tap number to provide characteristics shown in FIGS. 65 and 66 is normally about 16N (N is number of bandwidth division) and 80N for that in FIG. 61.

These are detailed descriptions of bandwidth division Kalman equalizer BDKEQ shown in FIGS. 60 and 64. Henceforth construction methods of band pass filters FILN(iJ) (i=0, 1 and 2; J=1 to N) are dealt with. The construction of FILN(iJ) (i=0, 1 and 2; J=1 to N) is given as shown in FIG. 67. The configuration in FIG. 67 is transversal filter of L taps (L is a certain integer) where T is of course delay element of T second and F_(iJl) 's (i=0, 1 and 2; J=1 to N; l=1 to L) are l'th tap weight coefficient in the band pass filter of FILN(iJ). Parts of F_(iJl) 's in FIG. 67 provide the values of their input signals multiplied by the amount of f_(iJl) which is value of tap weight of F_(iJl) at their output. SUM is of course concentrator. For the convenience of description, input terminal of FILN(iJ) is denoted as g and output terminal as g_(iJ). L, as mentioned before, must be chosen to be about 80N for FILN(0J)'s and about 16N for FILN(iJ)'s (i=1 and 2). Tap weights f_(0Jl) (J=1 to N; l=1 to L) for filter FILN(0J) with rectangular frequency gain characteristics used in the configuration of FIG. 60 is given by following Eq. (27). ##EQU11## The character at the right shoulder of exponential symbol e denotes imaginary number unit and not number of band pass filter. The value of A must be chosen to be L/2 or the number nearest to that value. Next, one example of values of f_(iJl) (i=1 and 2; J=1 to N; l=1 to L) are given in following Eqs. (28) and (29). ##EQU12## The notation of j at the right shoulder of exponential symbol e is explained above. FILN(iJ) (i=1 and 2; J=1 to N) obtained by the use of these coefficients have gentle frequency gain characteristics of raised cosine type shown in FIGS. 65 and 66. The first series filters FILN(1J)'s and second series ones FILN(2J)'s mutually compensate their gain characteristics at their common pass bands resulting in flat transmission gain characteristics by addition by mutual addition. Synthesized transmission function of these filters is of course that of through (direct connection) circuit. The band pass filters with tap weights given by Eqs. (28) and (29) can be applied to the bandwidth division Kalman equalizer of the type of FIG. 64. These are explanation of composing method of FILN(iJ) (i=1 and 2; J=1 to N).

Above illustration is concerned with bandwidth Kalman equalizer making use of Kalman equalizers with NT tap spacing each of which is related to a pass band provided by one of FILN(iJ) (i=0, 1 and 2; J=1 to N) filters. These equalizers working on pieces of partial bandwidth which amount to be N pieces in the composition of FIG. 60 and 2N in the composition of FIG. 64 are called partial equalizers. Only necessities concerning partial equalizers are that they must have NT second of tap spacing and can deal with complex numbers and hence conventional complex number learn and identification equalizer may have qualifications for partial equalizer. Because of low convergence ability, this kind of equalizer will not be treated in this invention. Nevertheless, the bandwidth division fast Kalman equalizer BDKEQ which utilizes complex number fast Kalman equalizer composing a part of this invention and formerly dealt with minutely for its composition and action, is very useful system for the reason of good convergence characteristics and small amount of calculating labor. The composition of BDKEQ which uses band pass filters of the type shown in FIGS. 65 and 66 is illustrated in FIG. 68. The composition of FIG. 68 can be obtained from composition of bandwidth division Kalman equalizer BDKEQ shown in FIG. 64 by replacing each (iJ) Kalman equalizer with NT tap spacing KEQQ(iJ) with (iJ) fast Kalman equalizer with NT tap spacing FKEQQ(iJ). Each (iJ) fast Kalman equalizer with NT tap spacing FKEQQ(iJ) is identical with each other and can be realized by complex number fast Kalman equalizer FKEQQ shown in FIGS. 26 and 27 with modification of change from original T second to NT second of tap spacing at complex tapped delay lines TDLQ₁ and TDLQ₂ involved in complex number transmission function TFQ and complex number fast Kalman controller FKCQ respectively and also change of delay, from original T second to NT second, of complex number delay elements DLQ₁ and DLQ₂ and consequent reduction of tap number. Each component and point in FIG. 68 are the same ones with the same notations in FIG. 64.

The explanation so far made in this invention will be summarized as follows. The role of automatic equalizers of real number and complex number types is illustrated first and consecutively conventional technics such as real number Kalman equalizer, complex number Kalman equalizer and real number fast Kalman equalizer are dealt with. The illustrations of new findings composing this invention are given following these explanations. Complex number fast Kalman equalizer and bandwidth division Kalman equalizer obtained by application of technic of bandwidth division to conventional complex number Kalman equalizer are treated in this order. Finally bandwidth division fast Kalman equalizer introduced by application of technic of bandwidth division to newly introduced complex number fast Kalman equalizer is dealt with.

These equalizers mentioned above, new and conventional included, work satisfactory, in relation to prescribed tap number, after their convergence and there are no distinctions in performances. The ratios of attained convergence and calculation labor especially multiplication amount are, however, very different among these equalizers. Necessary multiplication numbers until convergence are given as 10M (M is equalizer tap number) for fast Kalman type of equalizers and as 2N² for Kalman type of equalizers. This figure can be applied for both non bandwidth division types and bandwidth division types alike. The bandwidth division type equalizers, having tap number of one N'th (n is bandwidth division number) compared with that of non bandwidth division type equalizers, provide convergence speed N times faster than that of non bandwidth division types. The comparison of multiplication times, taking above mentioned two phenomena and also the multiplication labor for operation of band pass filters into account, shows that the ratios between them are 1:1/2:1/3:1/4 for Kalman, fast Kalman, bandwidth Kalman and bandwidth fast Kalman equalizers respectively, at an appropriate condition of 31 for tap number M and 16 for bandwidth division number N. This result shows systems consisting this invention provide superior operation to operation obtained from conventional equalizers. As shown minutely in this invention, equalizers proposed in this invention provide higher convergence performance in conjunction with necessary multiplication number and usefully operate at equalization of communication circuit.

From the foregoing, it will now be apparent that a new and improved Kalman equalizer has been found. It should be understood of course that the embodiments disclosed are merely illustrative and are not intended to limit the scope of the invention. Reference should be made to the appended claims, therefore, rather than the specification as indicating the scope of the invention. 

What is claimed is:
 1. A fast Kalman equalizer (FIG. 27) having a complex impulse response for equalizing a complex input signal comprising:(a) an input terminal Q₅ for receiving an input signal g_(k), (b) a transversal filter (TFQ) coupled with said input terminal Q₅, said transversal filter (TFQ) having at least a tapped delay line (TDLQ₁) with a plurality of adjustable taps and an adder (SUMQ₁) for summing up all the tap outputs of said tapped delay line (TDLQ₁), (c) an output terminal Q₇ coupled with the output of said adder (SUMQ₁) for providing an equalized output signal, (d) a complex number adder (ADQ₁) for providing e_(k) (Eq. 11), (e) a complex number A register controller (PTAQ) for providing k_(k) (Eq. 14), A_(k) (Eq. 18'), and q_(k) (Eq. 18), (f) a tap gain memory (RQ) for providing a tap coefficient to said tapped delay line (TDLQ₁), by accepting the increment of the tap weight using the formula Δh_(k) =k_(k) e_(k) (Eq. 12), (g) a multiplicator (MQ₂₂) for providing said product (Δh_(k) =k_(k) e_(k) (Eq. 12)) of the reference signal e_(k) from a reference terminal (Q₆) and k_(k) (Eq. 14), (h) a complex number dimension increaser (DIQ₁) for providing x_(k) (Eq. 9), (i) an S-register control (PTSQ) for providing (a) v_(k) which is the complex conjugate of v_(k) by the complex conjugate generator (C₂), said v_(k) is obtained by Eq. 19 (v_(k) =g_(k) +A_(k) ^(T) ·x_(k), where A_(k) ^(T) is the transposed matrix of the matrix A_(k)), and (b) S_(k) =S_(k-1) +v_(k) q_(k) (Eq. 20'), according to the values A_(k), x_(k), and g_(k), where v_(k) is the complex conjugate of v_(k), (j) a D-register control (PTDQ) for providing D_(k) (Eq. 25), m_(k) (Eq. 22 and Eq. 21), and w_(k) (Eq. 23), (k) a k-register control (PTkQ) for providing k_(k+1) by the formula: k_(k+1) =m_(k) -D_(k) w_(k) (Eq. 26), and (l) means for providing the value k_(k+1) obtained in said item (k) to the multiplicator (MQ₂₂) of the item (g), the A-register control (PTAQ) of the item (e), and the D-register control (PTDQ) of the item (j), for the next step calculation.
 2. A Kalman equalizer (FIG. 60, FIG. 64) having a complex impulse response for equalizing a complex input signal comprising:(a) an input terminal (Q₁ ') for receiving an input signal to be equalized, (b) a plurality of bandpass filters including input signal filters for dividing frequency bandwidth of the input signal and reference signal filters for dividing frequency bandwidth of a predetermined reference signal from a terminal (Q₆), (c) a plurality of partial equalizers (KEQQN) with adjustable tapped delay lines each coupled with a respective input signal filter and reference signal filter, for equalizing each divided frequency band of the input signal (d) an adder (SUMT) for summing up the outputs of said partial equalizers, (e) an output terminal (Q₇) coupled with the output of said adder (SUMT) for providing an equalized output signal, and (f) said partial equalizer being implemented by one selected from a Kalman equalizer (FIG. 8) and a fast Kalman equalizer (FIG. 27), a tap spacing in the partial equalizer being increased according to the number of said bandpass filters, and thus, the number of taps of said partial equalizer being reduced. 